Stuck on a Power Set Problem - What's the Lacking Set?

[nicnicman]

While practicing power set problems I came across one that has me stumped.

The problem asks: Is the following set is a power set of of a set?
{∅, {b, ∅}, {a}, {a, b}, {b}}

My answer: This set has 5 elements. Since 5 is not a power of 2, this cannot be the power set of any set.

The problem then asks: If the set is not a power set of a set, then what is the lacking set?

This is where I'm stuck. There are no similar problems in the book or on the Internet (that I can find), and I'm not sure how to answer this.

Thank you for any suggestions!

 

[SammyS]

While practicing power set problems I came across one that has me stumped.

The problem asks: Is the following set is a power set of of a set?
{∅, {b, ∅}, {a}, {a, b}, {b}}

My answer: This set has 5 elements. Since 5 is not a power of 2, this cannot be the power set of any set.

The problem then asks: If the set is not a power set of a set, then what is the lacking set?

This is where I'm stuck. There are no similar problems in the book or on the Internet (that I can find), and I'm not sure how to answer this.

Thank you for any suggestions!

If these are elements of the power set of some set, S, start my making a list of the elements of set S.

 

[nicnicman]

Okay if S = {∅, a, b},
then P(S) = [{∅, {∅}, {a}, {b}, {∅, a}, {∅, b}, {a, b}, {∅, a, b}]
Thus, the missing sets are {∅}, {∅, a}, and {∅, a, b}.

Is this correct?

 

[SammyS]

Okay if S = {∅, a, b},
then P(S) = [{∅, {∅}, {a}, {b}, {∅, a}, {∅, b}, {a, b}, {a, b}, {∅, a, b}]
Thus, the missing sets are {∅}, {∅, a}, and {∅, a, b}.

Is this correct?

Yes.

 

[nicnicman]

Great! Thanks for the help.